Saturation numbers for joins with graphs containing isolated vertices

Abstract

A graph $G$ is $H$-saturated if $G$ contains no $H$-copy as a subgraph, but adding any edge between two non-adjacent vertices in $G$ creates a copy of $H$. The saturation number $\mathrm{sat}(n,H)$ is the minimum number of edges in an $n$-vertex $H$-saturated graph. Cameron and Puleo [Discrete Math. 345 (2022), 112867] showed that $\mathrm{sat}(n,K_1\vee F)\le n-1+\mathrm{sat}(n-1,F)$ for all $n>|V(F)|$. It is natural to ask when the equality holds. Most known results on this bound consider non-empty graphs $F$ without isolated vertices. We study families in which $F$ contains isolated vertices. For $F=K_{p-1}\cup q{K}_1$, where $p\ge 4$ and $q\ge p-1$, we prove that $\mathrm{sat}(n,K_1\vee F)=(p-2)n-\left(\genfrac{}{}{0ex}{}{p-1}{2}\right)$ for $n\ge \max \{p+q,(\genfrac{}{}{0ex}{}{p}{2})\}$ and determine the unique extremal graph. We also give the exact result for $q=1$, where the extremal graph has a different structure. For $F=p{K}_2\cup q{K}_1$, in the natural range $n\ge 2p+q+1$, we prove that $\mathrm{sat}(n,K_1\vee F)=n+3p-4$ holds exactly when $n\ge 3p-2$, and the order condition is sharp. Furthermore, we characterize all extremal graphs.